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ACT Math : How to find the diameter of a sphere

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Example Question #4 : Spheres

If a sphere has a volume of $36\pi$ , what is its diameter?

Possible Answers:

$2\sqrt{3}$

$\sqrt{3}$

Correct answer:

Explanation:

1. Use the volume to find the radius:

$Volume=\frac{4}{3}\pi r^{3}$

$36\pi = \frac{4}{3}\pi r^{3}$

$\frac{3}{4}(36\pi) = (\frac{4}{3}\pi r^{3})\frac{3}{4}$

$27=r^{3}$

r=3

2. Use the radius to find the diameter:

$d=2r=2\cdot 3=6$

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Example Question #2 : Spheres

A sphere has a volume of $36\pi$ . What is its diameter?

Possible Answers:

144

Cannot be determined from the information given

$3\pi$

Correct answer:

Explanation:

This question relies on knowledge of the formula for volume of a sphere, which is as follows: $V=\frac{4}{3}\pi r^{3}}$

In this equation, we have two variables, and . Additionally, we know that $V=36\pi$ and is unknown. You can begin by rearranging the volume equation so it is solved for , then plug in and solve for :

Rearranged form:

$r=\sqrt[3]{\frac{3}{4\pi}V}$

Plug in $36\pi$ for V

$r=\sqrt[3]{\frac{3}{4\pi}36\pi}$

Simplify the part under the cubed root

1) Cancel the $\pi$ 's since they are in the numerator and denominator.

2) Simplify the fraction and the :

$36*\frac{3}{4}=27$

Thus we are left with

$r=\sqrt[3]{27}$

Then, either use your calculator and enter $27^{\frac{1}{3}}$ Or recall that $3^{3}=27$ in order to find that r=3 .

We're almost there, but we need to go a step further. Dodge the trap answer "" and carry on. Read the question carefully to see that we need the diameter, not the radius.

d=2r

d=2*3=6

is our final answer.

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Example Question #1 : Spheres

A spherical plastic ball has a diameter of $4\:in$ . What is the volume of the ball to the nearest cubic inch?

Possible Answers:

$44\:in^3$

$269\:in^3$

$34\:in^3$

$56\:in^3$

$22\:in^3$

Correct answer:

$34\:in^3$

Explanation:

To answer this question, we must calculate the volume of the ball using the equation for the volume of a sphere. The equation for the volume of a sphere is four-thirds multiplied by pi, which is then multiplied by the radius cubed. The equation can be written like this:

$V = \frac{4}{3}\pi r^{3}$

We are given the diameter of the sphere in the problem, which is $4\:in$ . To get the radius from the diameter, we divide the diameter by . So, for this data:

$radius=\frac{diameter}{2}=\frac{4}{2}=2$

We can then plug our newly found radius of two into the equation to find the volume. For this data:

$Volume = \frac{4}{3}\pi r^{3}=\frac{4}{3}\pi \cdot (2)^{3} = \frac{4}{3}\pi\cdot 8$

We then multiply $\frac{4}{3}$ by .

$\frac{4}{3}\pi\cdot 8=\frac{32}{3}\pi$

We finally substitute 3.14 for pi and multiply again to get our answer.

$\frac{32}{3}\pi = 33.5$

The question asked us to round to the nearest whole cubic inch. To do this, we round a number up one place if the last digit is a 5, 6, 7, 8, or 9, and we round it down if the last digit is a 1, 2, 3, or 4. Therefore:

$33.5\rightarrow34$

Therefore our answer is $34\:in^3$ .

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Example Question #7 : Spheres

A boulder breaks free on a slope and rolls downhill. It rolls for 355 complete revolutions before grinding to a halt. If the boulder has a volume of 1436 cubic feet, how far in feet did the boulder roll? (Assume the boulder doesn't lose mass to friction). Round $\pi$ to 3 significant digits. Round your final answer to the nearest integer.

Possible Answers:

15606ft

23280ft

9978ft

11747ft

11682ft

Correct answer:

15606ft

Explanation:

The formula for the volume of a sphere is:

$V = \frac{4}{3}\pi r^3$

To figure out how far the sphere rolled, we need to know the circumference, so we must first figure out radius. Solve the formula for volume in terms of radius:

$1436 = \frac{4}{3}\pi r^3$

$\frac{1077}{\pi} =r^3$

$r \approx 6.999932$

Since the answer asks us to round to the nearest integer, we are safe to round to at this point.

To find circumference, we now apply our circumference formula:

$C = 2r\pi = 14\pi$

If our boulder rolled 355 times, it covered that many times its own circumference.

$355\cdot 14\pi = 4970\pi \approx 15606$

Thus, our boulder rolled for 15606ft

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Example Question #1 : How To Find The Diameter Of A Sphere

Find the diameter of a sphere whose radius is .

Possible Answers:

2d^2

4d^2

Correct answer:

Explanation:

To solve, simply remember that diameter is twice the radius. Don't be fooled when the radius is an algebraic expression and incorporates the arbitrary constant . Thus,

$\textup{diameter}=2r=2*2d=4d$

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ACT Math : How to find the diameter of a sphere

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #4 : Spheres

Example Question #2 : Spheres

Example Question #1 : Spheres

Example Question #7 : Spheres

Example Question #1 : How To Find The Diameter Of A Sphere

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